Detailed Statistical Equilibrium
In order to define the number of atoms populating a particular energy level n,
all the transition to and from that level must be considered. The equilibrium is reached when the sum of the two processes is zero.
Let's consider all bound­bound transitions only (neglecting ionization and
recombination)
Nm
Nn
n
m (> n)
incoming
level n
outgoing
level n
Nm
m (<n)
Detailed Statistical Equilibrium(2)
Let's consider all bound­bound transitions only (neglecting ionization and
recombination) with m≠ n, and then let's define:
Rmn and Rnm = probability of a radiative transition per unit time from the
levels m  n and n  m respectively
Cmn and Cnm = probability of a collisional transition per unit time from the
levels m  n and n  m respectively
the equilibrium is reached when Nn is constant with time, namely
dN
−
= N n ∑m R nm  C nm  −
dt
OUTgoing
∑m N m R mn
 C mn  = 0
INcoming
set of statistical equations whose solutions provide the various Nn and then determine the Nn /Nm in case LTE cannot be applied
in case also recombination and ionization are considered, these equations need to be modified
Transitions in which the electron “exits” level n:
to a higher energy level mu
Rnm = Bnm U
Cnm = NpQnm Photo – excitation (absorption) [1]
Collisionalexcitation
[2]
to a lower energy level ml
Rnm = Anm +Bnm U
Cnm = NpQnm Spontaneous& stimulated emission (decay) [3+4]
Collisionaldecay
[5]
Trasitions in which the electron ''enters'' level n:
from a higher energy level mu
Spontaneous& stimulated emission (decay) [6+7]
Rmn = Amn + Bmn U
Collisionaldecay
[8]
C = N Q mn
p
mn
from a lower energy level ml
Rmn = Bmn U
Cmn = NpQmn Photo – excitation (absorption) [9]
Collisionalexcitation
[1 0]
N.B. Np numberdensity ofscattering particles, U =4I/c radiation field energy density,
Therefore we can now write the equation of statistical equilibrium,
balancing transitions to and from a certain level n
[14]
[3]
[ 25]
n−1

4


N n [ ∑m≠n B nm
I  ∑m=1 Anm N
p ∑m≠n Q nm ]
c
=
4
= ∑m≠n N m B mn
I  ∑mn N m A mn N p ∑m≠n N m Q mn
 
c

[ 810]
[6]
[79]
and this set requires to be solved together with radiative transfer (Iv is present here)


h nm
gm N n
h nm
dI 
=−
B mn N m 1 −
I  
Anm N n 
ds
c
gn Nm
4
Detailed Statistical Equilibrium (practical applications)
Let's consider a simple case: an atom has two energy levels only (m=1 and n=2). We want do determine the ration between their population N2 /N1
This is a typical case for electronic transitions (~ 1eV, visible band) in ions like C, N, O in the WIM around hot stars (free electrons are collisional partners) as well as rotational transitons in the hyperfine structure of HI (~ 0.001 eV) in WNM and CNM (HI and H2 are collisional partners) or molecular lines.
Statistical equations are much simpler:
4
4
N 2 B 21
I   A 21 N p Q 21  = N 1 B 12
I  N p Q 12 
c
c
4
B 12
I  N p Q 12
N2
c
then
=
N1
4
B 21
I   A 21 N p Q 21
c
here we need to evaluate Iv together with this equation
estimating Iv the radiation field may assume different values in various locations and at different bands (radio through X and  ­rays) For ions (CII, NII, OII,... CIII, NIII, OIII, ...) in the ISM at a distance r from a star with temperature T* and radius R* in the visual domain we have:
∗2
R
I  = B  T ∗ ⋅
= W ⋅B  T
2
4r
∗

where
R∗2
W =
4r 2
is the “dilution factor” (check whether  is required)
Collisional partners are free electrons, then Np = Ne , and replacing Iv we get:
N2
N1
4
B 12
I   N p Q 12
− h  / kT
−h  / kT ∗
g2
g 2 N e Q 21 / A 21 e
W e
−1−1
c
=
=
∗
g1
4
g 1 1 N e Q 21 / A 21  W e −h  / kT − 1−1
B 21
I   A 21 N p Q 21
c
and here the dilution factor plays the most relevant role.
consequences from Iv (visual band)
at large distances from any star (r ≫ R*) the radiation field in the optical band becomes progressively weaker (Ne Q21/A21 ≫ W [ 0]): N2
N1
=
W e
−1−1
g2
e−h  / kT
=
−1
−h  / kT
g 1  A21 / N e Q 21 1
1N e Q 21 / A 21 W e
−1
g 2 N e Q 21 / A 21 e
g1
−h  / kT
−h  / kT
∗
∗
and collisional transitions are the most likely excitation/decay process
N2
N1
=
g2
e −h  / kT
g 1 A 21 / N e Q 21 1
while
 
N2
N1
=
LTE
g2
g1
e −h  / kT
deviations from LTE for a given Nn are provided by bn = Nn / (Nn)LTE :
b2
1
=
b1
A 21 / N e Q 21 1

N2
N1
=
b2 N 2
 
b 1 N 1 LTE
in case radiative processes are negligible (Ne Q21 ≫ A21) then b1 = b2 and N2 / N1=(N2 / N1)LTE i.e. collisions dominate  thermal equilibrium holds
Applications: (visual band)
typical ISM parameters:
nm
gn
~ 1
2
4
T k ≈ 10 −10 K
then
3
Q 21 ≈ 10−7− 10−6 cm s −1
nm
−6
Q nm =8.6⋅10
while
T 1k / 2 g n
3
cm s −1
8
A 21 ≈ 10 s −1
therefore, the electron (collisional partner) density Ne may play an important role for transition whose radiative decay term A21 is small. For typical Ne  0.1 cm­3 (far from stars) / Ne  104 cm­3 (in nebulae)
A21
N e Q 21
14
10 −10
≈
Ne
then
b2
15
≫ 1

b1
≪ 1
   
N2
N1
≪
N2
N1
LTE
rare collisional excitations are immediately followed by a radiative decay, bringing back the ion to its fundamental state. Contrary to LTE, (nearly) all the ions are in the fundamental state
Applications (2): (visual band)
for forbidden transitions A21 is much smaller: lets consider the magnetic dipole transition in [OIII] (nebulio)
4
T k ≈ 10 K
;
A 21 = 2.1⋅10−2 s −1
[O III = 5007A ]
A 21 = 7.1⋅10−3 s −1
[O III  =4959A ]
3
Q 21 ≈ 10−7 cm s −1
A 21
N e Q 21
with
5
10
≈
≈ 10 −100
Ne
3
4
N e ≈ 10 −10 cm −3
;
b2
b1
≈ 0.1− 0.01
the upper level is populated with a rate of 1­10 % wrt the LTE.
Levels with longer lifetimes (small A21 , i.e. forbidden transions) are favoured w.r.t. those transitions with high probability of spontaneous decay.
?The smallest the electron density, the highest is the fraction of ion in the excited state (and then the probability of a radiative transition increases)?
Applications (3): (radio domain)
application to HI hyperfine transition (molecular transitions need a few adjustements). The radiation field is that of the CMB (R­J approximation), with W=1
2

−2
=
2
k
T

B
c2
A 21 kT B
−h  / kT
I = 2k T B
N2
N1
=
g2
g1
e
1

A 21
N p Q 21
N p Q 21 h 

A 21
kT B
N p Q 21 h 
The upper level (2) is significantly populated in cold (and relatively dense) regions, where ionization is negligible. Collisional parnters are either other HI atoms or H2 molecules.
When A21/(NP Q21) [kT / hv] << 1 namely A21/(NP Q21) << 1
we get the Boltzmann's equation (T.E.) Applications (4): (radio domain)
hv ≪ kTk, kTex , and the exponential can be written as Taylor's series:
T k x T B
E n −E m
T ex =
where kT ex =
1 x
Nm gn
ln
x=
A 21
N p Q 21
t rad
kT k
kT k
t coll
1
=
⋅A 21⋅
=
h  N p Q 21
h
t rad
 
Nn gm
where
−h  / kT k
1
1−e
=
=
B 21 4  / c I   A21
A 21
≈
t coll
1
=
N p Q 21
1 h
⋅
A 21 kT k
3
c
where B 21=
A21
3
8 h 
t rad T k  t coll T B
T ex =
t rad  t coll
t rad ≫ t coll  T ex ≈ T k
then in case collisional processes dominate over radiative transitions, then LTE holds and N n gn
=
Nm gm
e h
−
nm
/
kT
ex
Recombination and population of various states:
[omissis]
Nn in case of recombination:
gas optically thin at all lines
gas optically thick at high energies (ex. Ly­ photons cannot escape)
[...to be expanded...]
HI ­ line
HI atoms are in relatively cold regions → Ne is negligible → collisional partners are either other HI atoms or H2 molecules
The ground level (n=1, l=0, m=0, s=1/2, J= L+S =1/2) is split into two sublevels where e­ and p+ have parallel/antiparallel spin. The electron angular momentum J=L+S can combine in vector addition with the nuclear angular momentum I to form the total angular momentum of the system F=J+I, providing two states +1 (parallel) / 0 (antiparallel). The statistical weights are g=2F+1 and then.... the upper energy level has a degeneracy (3 possible combinations) and this statistically produces g2 : g1 = 3 : 1 (p. 75 Dopita)
The energy difference between the two hyperfine levels is hv = 5.9 10 ­6 eV
The radiative decay has A21 = 2.9 10 ­15 s­1, namely trad = 10 7 yr
The collisional decay in case of NH ~ 10 cm­3 and with Q21~ 10 ­10 cm­3 s­1 has tcoll = 300 yr and then it is favoured;
A 21 kT k
−4 T
x =
≈ 4⋅10
N p Q 21 h 
NH
In general x ≪ 1 given that T  103 K and NH ~ 1 – 10 cm­3
LTE holds and Tex=Tk (=Ts known as spin temperature) and the abundance ratio is
and ¾ of the HI is in the excited state.
N2
N1
=
g2
g1
e
−h HI / kT
≈ 3
The (radiative) emissivity for this HI transition is
HI
=
N2
4
A 21 h HI
≈
1.6⋅10
33
−
N HI
and with an optically thin cloud with size l, I ~ Sv v = HIl , in the R­J
domain we can derive the brightness temperature TB
2
c
= HI⋅l⋅ 2 ≈ 2.6⋅10−15 N HI⋅l
2k HI
HI
and N HI = NHI ∙ l is known as column density (measured in cm­2) which can be determined in case the cloud is optically thin, by measuring TB and l .
HI
In case the cloud is optically thick from the radiative transfer
T B = T ex 1− e
HI
−HI
 ≈ T ex HI
and then if HI is in LTE (Tex=Tk)
HI =
TB
HI
T ex
=
TB
HI
Tk
−15
= 2.6⋅10
N HI⋅l
Tk
HI – line absorption
In case there is a background synchrotron source and an intervening HI cloud
1) Both the source and the cloud are revealed:
T B1 = T Bsync⋅e
−HI
T k 1−e
−HI

2) The LOS exclude the background source but pick up the cloud emission
T B2 = T k 1−e
−HI
 ≈ 2.6⋅10−15 N HI⋅l
3) Again as in 1) but at a frequency slightly different (but not far) from vHI 2
2
T B1 = T Bsync ≈ T Bsync
.taking 1) – 3) and then – 2) 2
 T B = T B1−T B1 = T k −T Bsync 1 −e
 T B −T B2 =−T Bsync 1 −e
− HI
−HI


This allows to derive opacity and temperature. Absorption is easier to observe in case of a strong background source
HI – line (courtesy of Monica Orienti)
D alprincipio diindeterminazione:
 =
A 21
2
≈ 5⋅10−16 Hz 
La riga osservata puo’ essere molto allargata pereffetto Doppler da:
• M otitermici;
• M otiturbolenti;
• M otisistematicisu grande scala (Rotazione galattica);
Es. se T k ~67 K , v~1 km/s
 ~5 kHz
motitermici: T ~1 00 K
v ~1 km/s
 ~5 kHz
motisistematici:
v ~200 km/s
 ~1 M Hz
 D
HI
=
vr
c
HI – line (courtesy of Monica Orienti)
HI – line : the rotation curve in spirals from the neutral gas
The rotation curve can splitted into three regimes:
1) bulge: spherical model for mass distribution and  = (r) 2) thin disk: the surface density is (r)
3) at large distances from the GC, the mass can be considered as point
1 2 3
ag=
1)
if
2)
G M R 
2
R
 ≈ r −
=
G
R

2
R
∫0 4  r
2
r dr
v R  ≈ R 1− 
G R 4  r  r 
a g = 2 ∫0
dr
2
2 1/ 2
R
R −r 
then
M  R  =
R
∫0
R 
= v
R
~ R  = 0
2
R 
= v
R
2
2 R v 2 r r
2  r  r dr =
dr
∫
1
/
2
0
2
2
G
R −r 
calculus omissis ­> v independent of R HI – line : the rotation curve in spirals from the neutral gas
3)
ag=
G M gal
R2
R 
= v
R
2
 v R  ≈ R −1 / 2 Keplerian regime
Motion as seen in a rotating point:
r 
­­ decreasing function of r after its maximum wr at P*
­­ ma Main Lines in the ISM
Radiative decay (and then emission lines) from a state collisionally excited is among the possible cooling processes in the ISM
Heated gas → collisional excitation increases →
→ (line) emission increases and gas cools
Cooling gas → collisional excitation decreases →
→ (line) emission decreases
Again, considering a 2­state atom, when collisions prevail:
N2
g2
Q 12
e −h  / kT
1
N1
=
g 1  A 21 / N e Q 21 1
=
Q 21  A 21 / N e Q 21 1
since terms related to radiation Iv have been neglected.
As previosly seen, A21 and Q21 depend on the transition parameters and the balance is ruled by Ne LTE / non­LTE is determined whether A21 is either ≫ or≪ than NeQ21 Let's define the critical density Necr
cr
e
N
=
A 21
Q 21
namely
A 21
cr
e
N Q 21
=1
Critical density:
It is characteristic of atomic species (and transitions)
It has a weak dependance on T (via Q ~ T ­1/2)
radiative decay (and then emission lines) from a state collisionally excited